a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which...

more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case. There...

by a Hamiltonian over a Poisson manifold. In 1973, Yoichiro Nambu suggested a generalization involving Nambu–Poisson manifolds with more than one Hamiltonian...

derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra...

of functions. If the odd Poisson bi-vector π i j {\displaystyle \pi ^{ij}} is invertible, one has an odd symplectic manifold. In that case, there exists...

mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure...

on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions f and g on the manifold is itself a Hamiltonian...

be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (a functor), Weyl's...

superalgebra. Every symplectic supermanifold is a Poisson supermanifold but not vice versa. Poisson manifold Poisson algebra Noncommutative geometry v t e...

of a Poisson manifold. A multisymplectic manifold of degree k is a manifold equipped with a closed nondegenerate k-form. A polysymplectic manifold is a...

Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation...

algebroid of a Poisson manifold there is a Lie algebroid structure on A* induced by this Poisson structure. Analogous to the Poisson manifold case one can...

almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost...

that defines the Poisson bracket. A closely related type of manifold is a contact manifold. A combinatorial manifold is a kind of manifold which is discretization...

arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to...

limit Phase space Symplectic manifold Liouville's theorem (Hamiltonian) Poisson bracket Poisson algebra Poisson manifold Antibracket algebra Hamiltonian...

Poisson boundary Poisson bracket, see Hamiltonian mechanics header Poisson games Poisson manifold Poisson ring Poisson supermanifold Poisson–Charlier polynomials...

be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (a functor), Weyl's...

gives altogether four different classes of constraints. Consider a Poisson manifold M with a smooth Hamiltonian over it (for field theories, M would be...

Symplectic manifold Symplectic structure Symplectomorphism Contact structure Contact geometry Hamiltonian system Sasakian manifold Poisson manifold Möbius...

his results is a formal deformation quantization that holds for any Poisson manifold. He also introduced the Kontsevich integral, a topological invariant...

algebra. The differential forms on a Poisson manifold form a Gerstenhaber algebra. The multivector fields on a manifold form a Gerstenhaber algebra using...

coordinates in which the Poisson bivector is constant (plain flat Poisson brackets). For the general formula on arbitrary Poisson manifolds, cf. the Kontsevich...

Daniel Sternheimer, Lichnerowicz formulated the first definitions of a Poisson manifold in terms of a bivector, the counterpart of a (symplectic) differential...

Novikov ring Poisson 1.   2.  Poisson algebra. 3.  A Poisson manifold generalizes a symplectic manifold. 4.  A Poisson–Lie group, a Poisson manifold that also...

algebra of operators on a Hilbert space has the Poisson algebra of functions on a symplectic manifold as a singular limit, and properties of the non-commutative...

{\displaystyle \mathbb {R} ^{2n}} , equipped with its Poisson bracket (with a generalization to symplectic manifolds, described below). It is a special case of the...

lowering of indices. In certain specialized applications, such as on Poisson manifolds, the relationship may fail to be an isomorphism at singular points...

The symplectic structure induces a Poisson bracket. The Poisson bracket gives the space of functions on the manifold the structure of a Lie algebra. If...

{\displaystyle T^{*}M} associated to Poisson manifolds ( M , π ) {\displaystyle (M,\pi )} are transitive if and only if the Poisson structure π {\displaystyle \pi...